While the cake cools, pour the milk into a medium-sized saucepan set over medium heat to simmer.
2¼ cup milk
While the milk is heating up, place the egg yolks, sugar, and cornstarch in a medium-sized bowl. Whisk well until combined and smooth.
7 large egg yolks, ¾ cup granulated sugar, 5 tablespoons cornstarch
Once the milk has begun to simmer, remove from the heat. Slowly stream half of the heated milk into the egg yolk mixture while whisking constantly to avoid cooking the yolks.
Once half of the milk is whisked in and smooth, stream the mixture back into the saucepan with the remaining milk, again, whisking well.
Place the pan back over medium heat, whisking constantly, until the mixture thickens and you see a few large bubbles pop on the surface. Boil for 30-60 seconds while whisking vigorously, then remove from heat.
Add the vanilla bean paste and butter and whisk to combine. Set aside.
1½ tablespoons vanilla bean paste, 1½ tablespoons unsalted butter
Use the handle of a wooden spoon or a skewer to poke evenly-spaced holes into the top of the cooled cake.
Pour the pastry cream over the surface of the cake and spread to all of the edges. Gently press the pastry cream down onto the cake so that it goes down into the holes that have been poked into the cake.
Cover the cake with plastic wrap pressed to the pudding to prevent a skin from forming. Place the pan into the refrigerator to chill for at least 4 hours before topping with ganache.